Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Borel cardinalities below

Author: Michael Ray Oliver
Journal: Proc. Amer. Math. Soc. 134 (2006), 2419-2425
MSC (2000): Primary 03E15; Secondary 37A20
Posted: March 14, 2006
MathSciNet review: 2213716
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Borel cardinality of the quotient of the power set of the natural numbers by the ideal $\mathcal{Z}_0$ of asymptotically zero-density sets is shown to be the same as that of the equivalence relation induced by the classical Banach space . We also show that a large collection of ideals introduced by Louveau and Velickovic, with pairwise incomparable Borel cardinality, are all Borel reducible to . This refutes a conjecture of Hjorth and has facilitated further work by Farah.

References

[AK00] Scot Adams and Alexander S. Kechris, Linear algebraic groups and countable Borel equivalence relations, J. Amer. Math. Soc. 13 (2000), no. 4, 909-943. MR 1775739 (2001g:03086)
[Far01] Ilijas Farah, Basis problem for turbulent actions. II. $c\sb 0$ -equalities, Proc. London Math. Soc. (3) 82 (2001), 1-30. MR 1794255 (2002c:03075)
[Hjo00] Greg Hjorth, Actions by the classical Banach spaces, J. Symb. Logic 65 (2000), no. 1, 392-420. MR 1782128 (2001h:03088)
[HK97] Greg Hjorth and Alexander S. Kechris, New dichotomies for Borel equivalence relations, Bull. Symb. Logic 3 (1997), no. 3, 329-346. MR 1476761 (98m:03101)
[JK84] Winfried Just and Adam Krawczyk, On certain Boolean algebras $\mathscr{P}{\omega}/{I}$ , Trans. Amer. Math. Soc. 285 (1984), no. 1, 411-429. MR 0748847 (86f:04003)
[KL97] Alexander S. Kechris and Alain Louveau, The classification of hypersmooth Borel equivalence relations, J. Amer. Math. Soc. 10 (1997), no. 1, 215-242. MR 1396895 (97e:03067)
[LV94] Alain Louveau and Boban Velickovic, A note on Borel equivalence relations, Proc. Amer. Math. Soc. 120 (1994), no. 1, 255-259. MR 1169042 (94f:54076)
[Maz00] Krzysztof Mazur, A modification of Louveau and Velickovic's construction for ${F}\sb \sigma$ -ideals, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1475-1479. MR 1626442 (2000j:03067)
[Sol96] S $\l$ awomir Solecki, Analytic ideals, Bull. Symb. Logic 2 (1996), no. 3, 339-348. MR 1416872 (97i:04002)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 03E15, 37A20

Retrieve articles in all journals with MSC (2000): 03E15, 37A20

Additional Information

Michael Ray Oliver
Affiliation: Department of Mathematics, University of California, Box 951555, Los Angeles, California 90095--1555
Address at time of publication: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3
Email: oliver@cs.ucla.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08207-4
PII: S 0002-9939(06)08207-4
Keywords: Borel equivalence relations
Received by editor(s): April 5, 2004
Received by editor(s) in revised form: October 25, 2004 and February 18, 2005
Posted: March 14, 2006
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|